Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Definition 2.4.3.5 (The Homotopy Coherent Nerve). Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. We let $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ denote the simplicial set given by the construction

\[ ([n] \in \operatorname{{\bf \Delta }}^{\operatorname{op}}) \mapsto \operatorname{Hom}_{\operatorname{Cat_{\Delta }}}( \operatorname{Path}[n]_{\bullet }, \operatorname{\mathcal{C}}_{\bullet } ) = \{ \text{Simplicial functors $\operatorname{Path}[n]_{\bullet } \rightarrow \operatorname{\mathcal{C}}_{\bullet }$} \} . \]

We will refer to $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ as the homotopy coherent nerve of the simplicial category $\operatorname{\mathcal{C}}_{\bullet }$.